Let Q be a simple algebraic group of type A or C over a field of good positive characteristic. Let x ∈ q = Lie(Q) and consider the centraliser q_x = {y ∈ q: [xy] = 0}. We show that the invariant algebra S(qx)~(qx) is generated by the pth power subalgebra and the mod p reduction of the characteristic zero invariant algebra. The latter algebra is known to be polynomial [17] and we show that it remains so after reduction. Using a theory of symmetrisation in positive characteristic we prove the analogue of this result in the enveloping algebra, where the p-centre plays the role of the pth power subalgebra. In Zassenhaus' foundational work [30], the invariant theory and representation theory of modular Lie algebras were shown to be explicitly intertwined. We exploit his theory to give a precise upper bound for the dimensions of simple qx-modules. An application to the geometry of the Zassenhaus variety is given. When g is of type A and g = t ? p is a symmetric decomposition of orthogonal type we use similar methods to show that for every nilpotent e∈t the invariant algebra S(pe)~(t_e) is generated by the pth power subalgebra and S(pe)~(K_е) which is also shown to be polynomial.
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